4 Many-body inputs

The starting point in the study of large amplitude collective dynamics is the identification of the degrees of freedom to go into the equations of motion. Although collective coordinates are not direct observables, they are treated as physical degrees of freedom; they are required for the construction of the PES as well as the associated inertia- and dissipation tensors. To achieve a satisfactory description of fission observables, such as the fragment mass distribution or the total fragment kinetic energy, it is essential to include a sufficiently rich set of collective coordinates. For example, even though the principal fission degree of freedom is the overall elongation, it is necessary to also include a shape coordinate breaking reflection symmetry to obtain realistic fragment mass yields. It was argued long ago Nix (1969) that a reasonable description must use a minimum of five degrees of freedom, namely overall elongation, necking, reflection asymmetry, and the shapes of the two emerging fragments. It appears that an overall intensity of pairing correlations, treated as a degree of freedom, should also be added to this list Staszczak et al. (1989); Giuliani et al. (2014); Sadhukhan et al. (2014); Zhao et al. (2016); Bernard et al. (2019). However, the number used in actual studies is often smaller, primarily due to computational considerations.

Within the framework of MM treatments, the principal collective degrees of freedom are those characterising the nuclear shape. A variety of shape families have been employed. Probably the most widely used are the three-quadratic-surface parameterisation in Nix (1969), and the parameterisation of Brack et al. (1972), which have three parameters. A detailed discussion of the advantage of one particular parameterisation over another can be found in Möller et al. (2009). However, even five shape degrees of freedom may not always be sufficient. For example, triaxial shape deformations are often important in the region of the first saddle.

Self-consistent treatments based on nuclear energy density functionals have used multipole moments of the matter distribution as constraints to play the role of collective coordinates Krappe and Pomorski (2012); Younes et al. (2019). The primary collective coordinates employed in such studies are the quadrupole moments $Q_{20}$ and $Q_{22}$ used to control the overall distortion and triaxiality of the system, respectively, the octupole moment $Q_{30}$, used to control its reflection asymmetry, and the neck parameter or the hexadecapole moment $Q_{40}$. An interesting possibility is to generalise the use of a set of multipole moments as the constraining operators by using the density distribution itself, see section 3.2.2 for further discussion.

It is important to recognise the principal difference between the use of the nuclear shape as a (multi-dimensional) collective variable, as is done in the macroscopic-microscopic approaches, and the use of a set of density moments, as is being done in the microscopic treatments. Whereas the former approach calculates the properties of the system having the specified shape, the latter automatically performs energy minimisation so the system being treated is the one having the lowest energy subject to the specified moment constraints. Consequently its shape (or more generally: its matter distribution) is not under complete control. As discussed in section 3.2.1, the self-consistent density distribution may exhibit discontinuities as the moments are varied smoothly as a small change in the constraints might cause the new minimal state to have a quite different spatial appearance. This problem is particularly severe near the scission point, where there might be a major reorganisation of orbital fillings. A recent detailed study of this problem Dubray and Regnier (2012) developed diagnostic tools for identifying its presence and demonstrated how additional constraints could help. In any case, no set of collective coordinates were found that could eliminate the problem entirely. It is therefore clear that at least three collective constraints are needed to mitigate such discontinuities.

Fluctuations of the pairing field have also been used as collective coordinates Staszczak et al. (1989), see section 3.10. Here, a constraint on the dispersion in particle number $⟨N^2⟩-{⟨N⟩}^2$ is imposed to control the strength of the pairing field Vaquero et al. (2011, 2013). Studies of fission dynamics have shown that the coupling between shape and pairing degrees of freedom has in fact a significant effect on the collective inertia and, therefore, on the dynamical paths in the collective space. In particular, it may have a pronounced influence on spontaneous fission half-lives Sadhukhan et al. (2014); Zhao et al. (2016); Bernard et al. (2019). Pairing coordinates may also be important for the odd-even staggering in the fission yields Mirea (2014); Rodriguez-Guzmán and Robledo (2017). Induced fission is traditionally treated in a finite-temperature framework, where pairing is quickly quenched by the statistical fluctuations. Here, again, the dynamical treatment of pairing could substantially change the picture.

Generally, the introduction of additional collective coordinates increases the required numerical effort significantly. Nevertheless, for more refined descriptions, there is a need for a few additional collective variables that are not shape-related. One is the projection of total angular momentum on the fission axis, usually denoted by $K$ Nadtochy et al. (2012); Bertsch et al. (2018), which affects the angular distribution of the fission fragments. In a recent study, the configuration space was constructed in the HF approximation using the $K$-partition numbers as additional constraints Bertsch et al. (2018).

Another additional collective degree of freedom is related to the isospin. Except for TDHF, TDHFB, and DFT-Langevin, fission treatments have usually assumed that the fragments retain the same proton-to-neutron ratio as that of the mother nucleus. While some progress has recently been made in incorporating this degree of freedom into the MM treatments Möller et al. (2014); Möller and Ichikawa (2015); Möller and Schmitt (2017), further developments are still needed.

A near-term challenge will be to take advantage of newly available extensive computing resources and expand the space of collective coordinates, with the aim of obtaining a more realistic description of the evolution of the fissioning nucleus into fragments, especially in the region where nascent fragments appear near and beyond scission.

The ATDHFB (Adiabatic time-dependent HFB) and GCM+GOA formalisms are often applied to derive collective inertias for the CSE. In the ATDHFB this requires the inversion of the full linear response matrix. From a computational point of view, this is a daunting task that has been often alleviated by imposing various approximations Schunck and Robledo (2016). Typically, fission calculations rely on the ATDHFB inertias within the so-called non-perturbative cranking approximation, where the non-diagonal terms of the linear response matrix are neglected and the derivatives of the generalised density matrix with respect to the collective variables are computed numerically Baran et al. (2011). Very recently, both the exact and non-perturbative cranking GCM+GOA inertias have been computed for the first time Giuliani and Robledo (2018), showing that the non-perturbative cranking ATDHFB inertias can be reproduced even without the inclusion of collective momentum variables.

In the TDGCM framework, the expression for the collective kinetic energy can be obtained using either the GCM or the ATDHFB formalism. While the latter approach leads to the physical inertia in the case of translational motion Ring and Schuck (1980), the GCM approach may be incorrect if the conjugate collective variables are not included as collective degrees of freedom. This requires doubling the dimensionality of the collective space and in practice this is rarely if ever done Goeke and Reinhard (1980). To obtain a more realistic inertia, the ATDHFB expression is sometimes used in the GCM approaches. The possibility of using fully consistent GCM with pairs of collective variables would be desirable in the future.

Given the current status, several aspects should be addressed in order to reduce the source of uncertainties in the estimation of collective inertias. Among the most impelling ones is the calculation of the exact ATDHFB inertias. This is desirable because, according to the instanton formulation (see Skalski (2008) and section 3.6), it is the ATDHFB that provides a compatible framework to tackle the problem of nuclear dynamics under the barrier. The full linear response matrix has been inverted in Lechaftois et al. (2015) under some approximations but this method has not yet been extended to fission studies. Alternatively, one could try an approach along the lines of the Finite Amplitude Method (FAM) Hinohara (2015), where rather than inverting the linear response matrix itself one computes its action on the time derivative of the density matrix entering in the expression of the collective inertias. Such an approach has already been proposed and tried long time ago Dobaczewski and Skalski (1981), and, as advocated in Dobaczewski (2019), the time is ripe to start implementing it routinely in all ATDHFB calculations. Regardless of the practical implementation, the estimation of the exact ATDHFB inertias is a crucial step to understand the validity of the non-perturbative cranking approximations, which will reduce the uncertainties related to the collective inertias and bring a sounder estimation of collective kinetic energies and in the general adiabatic description of the fission process.

When it comes to non-adiabatic formulations, collective inertia can be derived within the DDD formalism Mirea (2019). For low collective velocities, the DDD inertia reduce to the cranking expressions.

In most treatments of the fission dynamics based on microscopic theory, it has been assumed that the collective degrees of freedom are well decoupled from the intrinsic degrees of freedom, usually referred to as the adiabatic assumption. Unfortunately, the nuclear $A$-body wave function of the nucleus cannot, in general, be expressed in terms of slow and fast components. Indeed, the typical time scale of nuclear collective modes is only slightly greater than the single-particle time scale Nazarewicz (2001). In the context of fission, the adiabatic approximation is questionable as the collective motion is highly dissipative Blocki et al. (1978); Bulgac, Jin and Stetcu (2019), see section 3.5.

There is therefore an urgent need for addressing the collective dissipation within a microscopic framework. While this presents a significant computational undertaking, the most immediate task consists in deriving the appropriate expressions for the dissipation in the particular microscopic model employed, a problem that is still quite unsettled Barrett et al. (1978). Another challenge is to identify high-quality fission data that will constraint the dissipation tensor.

The nuclear level density is a key ingredient of the Hauser-Feschbach statistical theory of nuclear reactions. Modeling many aspects of fission reactions rely on this type of statistical reaction theory: a first example is nucleon-induced fission, where the capture of the projectile by the target and the fission of the resulting compound nucleus are treated as a two-step process. Another example is the prompt de-excitation of the nascent fission fragments, which can be treated as compound nuclei undergoing statistical decays. Especially for applications in nuclear astrophysics, such as the calculation of fission transmission coefficients and fission yields, reliable predictions of nuclear level densities over a broad range of excitations for a large region of nuclei are desired.

Three main classes of nuclear level density models exist: analytical models (such as the back-shifted Fermi gas), configuration-interaction methods, and combinatorial model. Due to their simplicity, analytical models are often used in reaction codes, but they do not account for specific nuclear structure effects to a satisfactory degree. While in principle very powerful, configuration-interaction shell-model methods have so far been applied only to a limited number of light and medium-mass nuclei because of their computational complexity. In contrast, the shell-model Monte Carlo (SMMC) approach is capable of calculating level densities of heavy nuclei and was applied to nuclei as heavy as the lanthanides Alhassid et al. (2008). SMMC level densities have the advantage that they include contributions from both intrinsic and collective excitations. However, application of SMMC across the nuclear chart will require large computational resources.

Combinatorial models do not suffer from this hurdle and have been applied on the scale of the nuclear chart. These are usually based on the microscopic single-particle levels (provided by DFT calculations, microscopic-macroscopic approaches, or analytical optical potentials) from which the many-quasiparticle excited states are obtained after pairing has been included. The level density obtained by such combinatorial counting must be augmented by the effect of excited states that are mostly collective in nature. Most important is the appearance of rotational bands for deformed nuclei which may increase the level density by more than an order of magnitude even at moderate excitation energies. Even though this effect is very important, most treatments have long included it only by means of an empirical formula based on the moment of inertia of the nucleus Bjørnholm et al. (1973). However, more recent approaches have considered each individual many-quasiparticle excited state to be a rotational bandhead Uhrenholt et al. (2013), thus avoiding the introduction of adjustable parameters. Collective vibrations have also been included Hilaire et al. (2012); Uhrenholt et al. (2013), but these are most often neglected as they have been found to have only a small impact at low excitations compared to the rotational enhancement.

An additional aspect of the modelling of these collective enhancements is their dependence on the nuclear shape. For example, photofission rates are sensitive to the ratio of level densities at the ground state and at the fission saddle point. Many transport models use the level density (as a function of the collective coordinates) to relate the local excitation energy to a local temperature. Furthermore, recent transport treatments of fission have employed shape-dependent level densities to guide the nuclear shape evolution Ward et al. (2017), an approach that automatically takes account of the gradual decrease of pairing and shell effects at increasing excitation. The effect of this energy dependence is often emulated by using a phenomenological damping function for the level density. Finally, shape-dependent microscopic level densities are also important for the division of the internal excitation energy between the pre-fragments at scission Albertsson et al. (2020).

Statistical quantities are important in many aspects of fission, and microscopic theory is needed to go beyond the current empirical modelling of their dependence on shape and other variables. Particularly challenging is the problem of calculating shape-dependent level densities with a proper description of the gradual erosion of the shell effects with increasing energy.